I have my Apache server running in the /srv/http/ directory. I changed the group of it to httpadmin and then added my user to that group. Then, I changed the permissions of that directory to rwxrwxr-x, which means everyone in the group httpadmin should be able to write in that directory, right?

check your identity with id | grep httpadmin || echo check permissions. command. If you have any output here, you should add yourself to this group. You can use pw, usermod or any same command and then login your system again.

Next point you should check is permissions which written files have. At my home server I use sticky group to doesn't matter about default group I write and permission umask I have.